## 81-piece Sudoku Jigsaw

For fans of Killer Sudoku, Samurai Sudoku and other variants

### 81-piece Sudoku Jigsaw

Most Sudoku Jigsaw puzzles have 9 pieces, corresponding to the 9 regions which are typically contiguous.

Here we have a couple of puzzles where the squares that form each region are completely disjoint. Squares in the same region have the same colour, eg. each pink square contains the digits 1 to 9.

81-piece Sudoku Jigsaws
Jigsaw-A+B.jpg (60.76 KiB) Viewed 591 times

Hint: The two puzzles shown here are in fact closely related. If you can deduce what that relationship is the puzzles will become much easier.

Mathimagics
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### 81-piece Sudoku Jigsaws (Print friendly)

Here are the same pair of related puzzles in a print-friendly format. Each square is labelled with a region (block) identifier (A, B ... I). The 9 squares labelled A are a block and so must contain the digits 1 ... 9. Same for B, C etc.

I note that while the color format above is obviously impractical for solving purposes, the print-friendly format here obscures the relationship between the two.

81pc Jigsaws A+B (print friendly)
Jigsaw-AB-PF.gif (29.74 KiB) Viewed 575 times

Mathimagics
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### Re: 81-piece Sudoku Jigsaw

Mathimagics

I am obviously missing something here. You say:
Hint: The two puzzles shown here are in fact closely related. If you can deduce what that relationship is the puzzles will become much easier

I loaded them both into JSudoku and was surprised to find they were both unique with only 11 givens. I knew you could improve on 17 with appropriate jigsaw structure but did not think you could get that low. However JSudoku could not even start to solve them.

So from my position if I could deduce the relationship they would become solvable not easier, but in any case I cannot see any relationship in the layout, clues or solutions.

Maurice
HATMAN

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### Re: 81-piece Sudoku Jigsaw

Maurice,

I have a question - if JSudoku can't solve them, how does it know they have unique solutions?

I looked for "JSudoku", found 2 different ones on SourceForge but they both appear to be standard Sudoku solvers written in Java, not Jigsaw Sudoku solvers, so I couldn't answer this question for myself. I think "Java Sudoku" and "Jigsaw Sudoku" have same common abbreviation so perhaps this is why I can't find it.

Meanwhile, it may surprise you you to know that, in the universe of completely arbitrary Jigsaw patterns, 8-clue unique puzzles are generally the rule, not the exception.

Cheers
Jim
Last edited by Mathimagics on Tue Feb 07, 2017 4:26 am, edited 1 time in total.

Mathimagics
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### Re: 81-piece Sudoku Jigsaw

A big hint for deducing the relationship ...

Hidden Text: Show
Pick a color (region code). Make a list of all 9 cells that have the same color (region code) in one puzzle, and their corresponding codes in the other puzzle.

Mathimagics
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Mathimagics wrote:
I looked for "JSudoku", found 2 different ones on SourceForge but they both appear to be standard Sudoku solvers written in Java, not Jigsaw Sudoku solvers

software from Jean-Christophe Godart

Pat

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### Re: 81-piece Sudoku Jigsaw

Thanks Pat,

After much trial and error I got it to work. The JSudoku specs for each puzzle are given below. Save each into a text file and load with FIle->Open. The solver produces the right answer in both cases.

Code: Select all
`SumoCueV1=0J3=0J0=9J5=0J4=0J7=0J8=0J6=3J2=0J1=0J8=0J2=2J6=0J5=0J3=0J7=0J4=0J1=0J0=0J0=0J1=0J3=6J2=0J5=0J4=0J7=0J8=0J6=0J6=0J3=0J8=8J7=0J2=0J0=0J1=0J5=0J4=0J2=5J4=0J7=0J1=0J0=0J6=0J8=0J3=6J5=0J5=0J7=1J1=0J8=0J6=0J2=0J0=0J4=0J3=0J7=0J8=0J4=0J6=0J1=0J3=0J5=0J0=0J2=2J1=0J6=0J2=0J0=0J4=0J5=0J3=0J7=0J8=0J4=0J5=0J0=0J3=9J8=0J1=0J2=7J6=0J7`

Code: Select all
`SumoCueV1=0J3=0J0=9J5=0J4=0J7=0J8=0J6=5J2=0J1=0J0=0J1=6J4=0J6=0J8=7J3=0J5=0J7=0J2=0J6=0J5=0J1=0J8=0J3=0J2=0J0=3J4=0J7=0J2=0J4=0J7=7J1=0J6=0J5=0J8=0J0=0J3=0J5=4J7=0J2=0J3=0J4=0J0=0J1=0J6=0J8=0J7=0J6=2J0=0J2=0J1=0J4=0J3=0J8=0J5=0J8=0J3=0J6=0J5=0J2=0J7=0J4=0J1=0J0=0J4=0J8=5J3=0J7=0J0=0J1=0J2=0J5=0J6=0J1=0J2=1J8=0J0=0J5=0J6=0J7=6J3=0J4`

I have to say that is one of the least attractive puzzle formats I have ever seen!

Mathimagics
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### Re: 81-piece Sudoku Jigsaw

Well, I just love a challenge and I couldn't let this one go.

I'll cut a long story short and just explain the results. What the hint means is that a fixed puzzle group code in one puzzle gives the 9 locations of a fixed digit in the other puzzle.

For example, if you look at Puzzle 1 you'll see that the 1 in r9c3 corresponds to a Group A entry in Puzzle 2. The hint then implies that the location of the other '1s in Puzzle 1 are the locations of the other A's in Puzzle 2.

Similar comments apply to finding the locations of the digits in Puzzle 2.

You'll find that in each puzzle there are 8 different digit clues that match a group, which gives you all except 1 non-clue digit, which must be placed in the only available cells left in the puzzle.

Simple, that only took about 7 hours to figure out. Here are the solutions to the puzzles :

Hidden Text: Show
Code: Select all
`Puzzle 1819473652356917428128594736683751294547216389972365148734628915265149873491832567Puzzle 2719256438182467953498673125325849617953721846541382769674935281267518394836194572`

You don't have to know anything about how to solve Jigsaw puzzles and you don't need any special software, just a lot of patience, or determination !

Leren
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### Re: 81-piece Sudoku Jigsaw

Leren wrote:You don't have to know anything about how to solve Jigsaw puzzles and you don't need any special software, just a lot of patience, or determination !

Amen! Congratulations Leren, have a cigar!! I am truly impressed.

What we actually have here are two Latin Squares with the property that each one acts as the jigsaw layout (aka GDL) for the other.

Thus all cells with the same color in one puzzle will have the same value in the other, and vice-versa. See the solution posted below.

This can only be done with pairs of Latin Squares that are orthogonal. Two Latin Squares L1, L2 are said to be orthogonal if every pair {L1(r,c), L2(r,c)} is unique. For size 9x9 it seems that roughly 1% of LS's have an orthogonal pair.

The concept of orthogonality can be extended to include GDL's (jigsaw layouts) - thus solving a Sudoku is simply finding a Latin Square that is orthogonal to the given GDL. (see note below)

My idea is to produce pairs of puzzles like this, in which the jigsaw codes are also obscured, but there are just enough givens and jigsaw codes to complete the puzzle.

---------------------------------------------------------------------------------------------------------------------
PS: for those of you familiar with Knuth's Dancing Links (DLX) algorithm, it's interesting to note that one of the exercises in Knuth Vol 4 (Section 7 - Combinatorial Searching, Exercise #17), asks how the problem of finding orthogonal pairs , for a given LS, can be expressed as an exact cover problem. The answer is exactly what DLX Sudoku solvers work with.
Last edited by Mathimagics on Wed Feb 08, 2017 9:00 am, edited 5 times in total.

Mathimagics
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### Re: 81-piece Sudoku Jigsaw

Here is the solution (the original givens are underlined):

Jigsaw-A+B-Soln.jpg (95.72 KiB) Viewed 524 times

Mathimagics
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### Re: 81-piece Sudoku Jigsaw

Mathimagics:

I'm sure you realize that your latest logo represents the number of solutions a puzzle is supposed to have.

What's purple and commutative?

What's green and very far away?

What's yellow and equivalent to the Axiom of Choice?

Bill Smythe
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### Re: 81-piece Sudoku Jigsaw

My granddaughter, while munching on Abelian grapes, said I should have an avatar, and that was the first 1 that came to mind ...

Mathimagics
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### Re: 81-piece Sudoku Jigsaw

Well done Leren

I tried to create a vanilla puzzle knowing the relationship but quickly realised they can only be trivial.

However a twin killer is possible (and I have not created one since about 2008).

This one is a just a proof of concept, so easy. If anyone is interested I'll do a couple of hard ones.

If you use JSudoku do not use jigsaw cages as they do not display disjoint groups well. Use extra groups which are coloured.

(The solutions are the same as above.)
HATMAN

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